In CRT type monitors there is a trick they used to call "Boost HT" back in the tube days,& which is also used with solid state circuitry.
On the Horizontal Output transformer,there is an overwind that produces a hundred or so volts at the horizontal frequency.
This is then rectified,filtered,& added to the normal power supply voltage to the Horiz & Vert outputs.
The filter capacitor is usually something like 10 to 20 uF at 400v ,& MUST be a low ESR cap--ordinary ones go bang after a few minutes!
The (half-wave) rectifier is a fast-recovery type --again,ordinary types go bang!
If the cap has become old & lost capacitance, the expected nice filtered dc is not smooth anymore,plus it's regulation is poor.
The result is the effect you have noted.
With the kind of display you are looking at,you wouldn't notice it,but the vertical scans will also be distorted.
Monitors used in industrial equipment are usually made by the same companies who make them for arcade games,etc.
If you look inside,you may find a manufacturer's name,& part number,& possibly you will be able to find a schematic on the Net.
if not,trace around the circuitry associated with the Horizontal Output transformer,& you should be able to find what you are looking for.
Take great care,as this area can BITE you!
You shouldn't need to do that part with the monitor energised,anyway!
If you have identified the circuitry,measure the "Boost HT" with a DMM,it will probably be something like 150volts dc.
If it is much less,you have probably found the fault.
If your DMM reads around 100v,so it almost looks OK,be suspicious,as some DMMs will give false readings with fairly high frequency rectified ,unfiltered dc.
The best thing to look at this with is an Oscilloscope.
If you have determined that this is the fault,change the capacitor (low ESR),& all will be well.
Good Luck!
VK6ZGO